Aleph 0

Aleph 0, written \(\aleph_0\) and said aleph null, is the cardinal corresponding to the cardinality of the natural numbers. As an initial (von Neumann) ordinal, it is considered the same as \(\omega\), while it may not be the same while in the absence of the axiom of choice.

In the context of AC, \(\aleph_0\) is the smallest infinite cardinal and, as such, larger than all numbers considered in googology. It can not be reached from below by any form of arithmetic, and as such may be considered analogous to an inaccessible.

Aleph 0 is also the cardinality of the integers, and of any infinite subset of the naturals or integers. Furthermore, Cantor proved, via diagonalization, that, surprisingly, aleph 0 is also the cardinality of the rational numbers.